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        <h1>DormandPrince</h1>
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          <div class="fullcommenttop" id="comment"><div class="comment cmt"><p>::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Given an unknown, time-dependent function y(t) governed by an Ordinary
Differential Equation (ODE) of the form y(t)' = f(t, y) where ' is d/dt,
compute y(t) using a (4,5)-order Dormand-Prince Integrator (DOPRI).  Note:
the integrateV method for a system of separable ODEs is mixed in from the
Integrator trait.</p></div><dl class="attributes block"> <dt>See also</dt><dd><span class="cmt"><p>http://adorio-research.org/wordpress/?p=6565
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      <p class="shortcomment cmt">Butcher tableau @see http://en.</p><div class="fullcomment"><div class="comment cmt"><p>Butcher tableau @see http://en.wikipedia.org/wiki/Dormand–Prince_method
</p></div></div>
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      <h4 class="signature">
      <span class="kind">val</span>
      <span class="symbol"> 
        <span class="name">c7</span>
        
        <span class="result">: <span name="scala.Double" class="extype">Double</span></span>
      </span>
      </h4>
      
    </li><li visbl="prt" name="scala.AnyRef#clone" data-isabs="false">
      <a id="clone():AnyRef"></a>
      <h4 class="signature">
      <span class="kind">def</span>
      <span class="symbol"> 
        <span class="name">clone</span>
        
        <span class="params">()</span><span class="result">: AnyRef</span>
      </span>
      </h4>
      <div class="fullcomment"><dl class="attributes block"> <dt>Attributes</dt><dd>protected[<a name="java.lang" class="extype" href="../../java/lang/package.html">lang</a>] </dd><dt>Definition Classes</dt><dd>AnyRef</dd><dt>Annotations</dt><dd>
                <span class="name">@throws</span><span class="args">()</span>
              
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      <a id="defaultStepSize:Double"></a>
      <h4 class="signature">
      <span class="kind">val</span>
      <span class="symbol"> 
        <span class="name">defaultStepSize</span>
        
        <span class="result">: <span name="scala.Double" class="extype">Double</span></span>
      </span>
      </h4>
      <p class="shortcomment cmt">The default step size for the t dimension
</p><div class="fullcomment"><div class="comment cmt"><p>The default step size for the t dimension
</p></div><dl class="attributes block"> <dt>Attributes</dt><dd>protected </dd><dt>Definition Classes</dt><dd><a name="scalation.dynamics.Integrator" class="extype" href="Integrator.html">Integrator</a></dd></dl></div>
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      <span class="kind">def</span>
      <span class="symbol"> 
        <span class="name">eq</span>
        
        <span class="params">(<span name="arg0">arg0: AnyRef</span>)</span><span class="result">: <span name="scala.Boolean" class="extype">Boolean</span></span>
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      <span class="kind">def</span>
      <span class="symbol"> 
        <span class="name">equals</span>
        
        <span class="params">(<span name="arg0">arg0: <span name="scala.Any" class="extype">Any</span></span>)</span><span class="result">: <span name="scala.Boolean" class="extype">Boolean</span></span>
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      <a id="error:Double"></a>
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      <span class="kind">var</span>
      <span class="symbol"> 
        <span class="name">error</span>
        
        <span class="result">: <span name="scala.Double" class="extype">Double</span></span>
      </span>
      </h4>
      <p class="shortcomment cmt">Estimate of the error in calculating y
</p><div class="fullcomment"><div class="comment cmt"><p>Estimate of the error in calculating y
</p></div><dl class="attributes block"> <dt>Attributes</dt><dd>protected </dd><dt>Definition Classes</dt><dd><a name="scalation.dynamics.Integrator" class="extype" href="Integrator.html">Integrator</a></dd></dl></div>
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      <a id="finalize():Unit"></a>
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      <span class="kind">def</span>
      <span class="symbol"> 
        <span class="name">finalize</span>
        
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                <span class="name">@throws</span><span class="args">()</span>
              
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      <p class="shortcomment cmt">:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Show the flaw by printing the error message.</p><div class="fullcomment"><div class="comment cmt"><p>:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Show the flaw by printing the error message.</p></div><dl class="paramcmts block"><dt class="param">method</dt><dd class="cmt"><p>the method where the error occured</p></dd><dt class="param">message</dt><dd class="cmt"><p>the error message
</p></dd></dl><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="scalation.util.Error" class="extype" href="../util/Error.html">Error</a></dd></dl></div>
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      <span class="kind">def</span>
      <span class="symbol"> 
        <span class="name">getError</span>
        
        <span class="result">: <span name="scala.Double" class="extype">Double</span></span>
      </span>
      </h4>
      <p class="shortcomment cmt">::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Get the error estimate.</p><div class="fullcomment"><div class="comment cmt"><p>::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Get the error estimate.
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="scalation.dynamics.Integrator" class="extype" href="Integrator.html">Integrator</a></dd></dl></div>
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      <a id="hashCode():Int"></a>
      <h4 class="signature">
      <span class="kind">def</span>
      <span class="symbol"> 
        <span class="name">hashCode</span>
        
        <span class="params">()</span><span class="result">: <span name="scala.Int" class="extype">Int</span></span>
      </span>
      </h4>
      <div class="fullcomment"><dl class="attributes block"> <dt>Definition Classes</dt><dd>AnyRef → Any</dd></dl></div>
    </li><li visbl="pub" name="scalation.dynamics.DormandPrince#integrate" data-isabs="false">
      <a id="integrate(Derivative,Double,Double,Double,Double):Double"></a>
      <h4 class="signature">
      <span class="kind">def</span>
      <span class="symbol"> 
        <span class="name">integrate</span>
        
        <span class="params">(<span name="f">f: Derivative</span>, <span name="y0">y0: <span name="scala.Double" class="extype">Double</span></span>, <span name="t">t: <span name="scala.Double" class="extype">Double</span></span>, <span name="t0">t0: <span name="scala.Double" class="extype">Double</span> = <span class="symbol">0.</span></span>, <span name="step">step: <span name="scala.Double" class="extype">Double</span> = <span class="symbol"><span class="name"><a href="Integrator.html#defaultStepSize:Double">defaultStepSize</a></span></span></span>)</span><span class="result">: <span name="scala.Double" class="extype">Double</span></span>
      </span>
      </h4>
      <p class="shortcomment cmt">::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Compute y(t) governed by a differential equation using numerical integration
of the derivative function f(t, y) using a (4,5)-order Dormand-Prince method to
return the value of y(t) at time t.</p><div class="fullcomment"><div class="comment cmt"><p>::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Compute y(t) governed by a differential equation using numerical integration
of the derivative function f(t, y) using a (4,5)-order Dormand-Prince method to
return the value of y(t) at time t.</p></div><dl class="paramcmts block"><dt class="param">f</dt><dd class="cmt"><p>the derivative function f(t, y)</p></dd><dt class="param">y0</dt><dd class="cmt"><p>value of the y-function at time t0, y0 = y(t0)</p></dd><dt class="param">t</dt><dd class="cmt"><p>the time value at which to compute y(t)</p></dd><dt class="param">t0</dt><dd class="cmt"><p>the initial time</p></dd><dt class="param">step</dt><dd class="cmt"><p>the middle step size
</p></dd></dl><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="scalation.dynamics.DormandPrince" class="extype" href="">DormandPrince</a> → <a name="scalation.dynamics.Integrator" class="extype" href="Integrator.html">Integrator</a></dd></dl></div>
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      <a id="integrate2(Derivative,Double,Double,Double,Double,Double,Double,Int):Double"></a>
      <h4 class="signature">
      <span class="kind">def</span>
      <span class="symbol"> 
        <span class="name">integrate2</span>
        
        <span class="params">(<span name="f">f: Derivative</span>, <span name="y0">y0: <span name="scala.Double" class="extype">Double</span></span>, <span name="t">t: <span name="scala.Double" class="extype">Double</span></span>, <span name="hmin">hmin: <span name="scala.Double" class="extype">Double</span></span>, <span name="hmax">hmax: <span name="scala.Double" class="extype">Double</span></span>, <span name="t0">t0: <span name="scala.Double" class="extype">Double</span> = <span class="symbol">0.</span></span>, <span name="tol">tol: <span name="scala.Double" class="extype">Double</span> = <span class="symbol">1E-5</span></span>, <span name="maxSteps">maxSteps: <span name="scala.Int" class="extype">Int</span> = <span class="symbol">1000</span></span>)</span><span class="result">: <span name="scala.Double" class="extype">Double</span></span>
      </span>
      </h4>
      <p class="shortcomment cmt">::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Compute y(t) governed by a differential equation using numerical integration
of the derivative function f(t, y) using a (4,5)-order Dormand-Prince method to
return the value of y(t) at time t.</p><div class="fullcomment"><div class="comment cmt"><p>::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Compute y(t) governed by a differential equation using numerical integration
of the derivative function f(t, y) using a (4,5)-order Dormand-Prince method to
return the value of y(t) at time t.  The method provides more customization
options.</p></div><dl class="paramcmts block"><dt class="param">f</dt><dd class="cmt"><p>the derivative function f(t, y)</p></dd><dt class="param">y0</dt><dd class="cmt"><p>value of the y-function at time t0, y0 = y(t0)</p></dd><dt class="param">t</dt><dd class="cmt"><p>the time value at which to compute y(t)</p></dd><dt class="param">hmin</dt><dd class="cmt"><p>the minimum step size</p></dd><dt class="param">hmax</dt><dd class="cmt"><p>the maximum step size</p></dd><dt class="param">t0</dt><dd class="cmt"><p>the initial time</p></dd><dt class="param">tol</dt><dd class="cmt"><p>the tolerance</p></dd><dt class="param">maxSteps</dt><dd class="cmt"><p>the maximum number of steps
</p></dd></dl></div>
    </li><li visbl="pub" name="scalation.dynamics.Integrator#integrateV" data-isabs="false">
      <a id="integrateV(Array[Derivative],VectorD,Double,Double,Double):VectorD"></a>
      <h4 class="signature">
      <span class="kind">def</span>
      <span class="symbol"> 
        <span class="name">integrateV</span>
        
        <span class="params">(<span name="f">f: <span name="scala.Array" class="extype">Array</span>[Derivative]</span>, <span name="y0">y0: VectorD</span>, <span name="t">t: <span name="scala.Double" class="extype">Double</span></span>, <span name="t0">t0: <span name="scala.Double" class="extype">Double</span> = <span class="symbol">0.</span></span>, <span name="step">step: <span name="scala.Double" class="extype">Double</span> = <span class="symbol"><span class="name"><a href="Integrator.html#defaultStepSize:Double">defaultStepSize</a></span></span></span>)</span><span class="result">: VectorD</span>
      </span>
      </h4>
      <p class="shortcomment cmt">::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Apply the integrate method to each derivative to compute the trajectory of
a time-dependent vector function y(t) governed by a separable system of
Ordinary Differential Equations (ODE's) where [f_j(t, y_j)] is an array of
derivative functions.</p><div class="fullcomment"><div class="comment cmt"><p>::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Apply the integrate method to each derivative to compute the trajectory of
a time-dependent vector function y(t) governed by a separable system of
Ordinary Differential Equations (ODE's) where [f_j(t, y_j)] is an array of
derivative functions.  Each derivative function takes a scalar t and a
scalar y_j = y(j).</p></div><dl class="paramcmts block"><dt class="param">f</dt><dd class="cmt"><p>the array of derivative functions [f_j(t, y_j)]</p></dd><dt class="param">y0</dt><dd class="cmt"><p>the initial value vector, y0 = y(t0)</p></dd><dt class="param">t</dt><dd class="cmt"><p>the time value at which to compute y(t)</p></dd><dt class="param">t0</dt><dd class="cmt"><p>the initial time</p></dd><dt class="param">step</dt><dd class="cmt"><p>the step size
</p></dd></dl><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="scalation.dynamics.Integrator" class="extype" href="Integrator.html">Integrator</a></dd></dl></div>
    </li><li visbl="pub" name="scalation.dynamics.DormandPrince#integrateVV" data-isabs="false">
      <a id="integrateVV(Array[DerivativeV],VectorD,Double,Double,Double):VectorD"></a>
      <h4 class="signature">
      <span class="kind">def</span>
      <span class="symbol"> 
        <span class="name">integrateVV</span>
        
        <span class="params">(<span name="f">f: <span name="scala.Array" class="extype">Array</span>[DerivativeV]</span>, <span name="y0">y0: VectorD</span>, <span name="t">t: <span name="scala.Double" class="extype">Double</span></span>, <span name="t0">t0: <span name="scala.Double" class="extype">Double</span> = <span class="symbol">0.</span></span>, <span name="step">step: <span name="scala.Double" class="extype">Double</span> = <span class="symbol"><span class="name"><a href="Integrator.html#defaultStepSize:Double">defaultStepSize</a></span></span></span>)</span><span class="result">: VectorD</span>
      </span>
      </h4>
      <p class="shortcomment cmt">::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Compute y(t), a vector, governed by a system of differential equations using
numerical integration of the derivative function f(t, y) using a (4,5)-order
Dormand-Prince method to return the value of y(t) at time t.</p><div class="fullcomment"><div class="comment cmt"><p>::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Compute y(t), a vector, governed by a system of differential equations using
numerical integration of the derivative function f(t, y) using a (4,5)-order
Dormand-Prince method to return the value of y(t) at time t.</p></div><dl class="paramcmts block"><dt class="param">f</dt><dd class="cmt"><p>the array of derivative functions [f(t, y)] where y is a vector</p></dd><dt class="param">y0</dt><dd class="cmt"><p>the value of the y-function at time t0, y0 = y(t0)</p></dd><dt class="param">t</dt><dd class="cmt"><p>the time value at which to compute y(t)</p></dd><dt class="param">t0</dt><dd class="cmt"><p>the initial time</p></dd><dt class="param">step</dt><dd class="cmt"><p>the step size
</p></dd></dl><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="scalation.dynamics.DormandPrince" class="extype" href="">DormandPrince</a> → <a name="scalation.dynamics.Integrator" class="extype" href="Integrator.html">Integrator</a></dd></dl></div>
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      <span class="symbol"> 
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        <span class="tparams">[<span name="T0">T0</span>]</span>
        <span class="result">: <span name="scala.Boolean" class="extype">Boolean</span></span>
      </span>
      </h4>
      <div class="fullcomment"><dl class="attributes block"> <dt>Attributes</dt><dd>final </dd><dt>Definition Classes</dt><dd>Any</dd></dl></div>
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      <a id="ne(AnyRef):Boolean"></a>
      <h4 class="signature">
      <span class="kind">def</span>
      <span class="symbol"> 
        <span class="name">ne</span>
        
        <span class="params">(<span name="arg0">arg0: AnyRef</span>)</span><span class="result">: <span name="scala.Boolean" class="extype">Boolean</span></span>
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      <div class="fullcomment"><dl class="attributes block"> <dt>Attributes</dt><dd>final </dd><dt>Definition Classes</dt><dd>AnyRef</dd></dl></div>
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        <span class="name">notify</span>
        
        <span class="params">()</span><span class="result">: <span name="scala.Unit" class="extype">Unit</span></span>
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